Fig. 1: Acoustic meta-grating for the generation of acoustic spatiotemporal vortex pulses.

a Schematics of the meta-grating generating a spatiotemporal vortex pulse for airborne sound by breaking the mirror symmetry. The period of the meta-grating is \(p=\) 33.34 mm. b The structure of the meta-grating, where the solid material is shown in gray, the position of the yellow block (air) is fixed, while the four white blocks (air) marked can be x-shifted to the left or right. The displacements of these blocks, \(\delta {x}_{i}\), break the mirror symmetry with respect to the x = 0 plane, and this asymmetry is quantified by the dimensionless parameter \(\eta\) (see explanations in the text). c–e Numerical simulations for the phase (top) and the amplitude (bottom) of the transmission spectrum function \(T({k}_{x},\omega )\) for different values of the asymmetry parameter \(\eta\). Phase singularities (vortices) with the winding numbers (topological charges) +1 and −1 are indicated by the white and black arrows, respectively. f The pair of vortices in the panels (d) and (e) corresponds to a single phase-singularity (nodal) line in the 3D space (k_x,ω,\(\eta\)) extended by the asymmetry parameter \(\eta\). The vortex pair emerges at the critical values of \({\eta }_{c}=0.40\). This nodal line and the separated vortices are topologically protected against small perturbations in the system.